Extended L-ensembles: a new representation for Determinantal Point Processes

TL;DR

This paper introduces extended L-ensembles, a unifying framework that encompasses all DPPs, simplifies likelihood calculations, and broadens the class of kernels usable for defining DPPs, including conditional positive definite kernels.

Contribution

The paper presents extended L-ensembles, proving they unify all DPPs, improve theoretical properties, and expand kernel options for defining DPPs.

Findings

All DPPs are extended L-ensembles and vice-versa.

Extended L-ensembles have simple likelihood functions.

Conditional positive definite kernels are suitable for defining DPPs.

Abstract

Determinantal point processes (DPPs) are a class of repulsive point processes, popular for their relative simplicity. They are traditionally defined via their marginal distributions, but a subset of DPPs called "L-ensembles" have tractable likelihoods and are thus particularly easy to work with. Indeed, in many applications, DPPs are more naturally defined based on the L-ensemble formulation rather than through the marginal kernel. The fact that not all DPPs are L-ensembles is unfortunate, but there is a unifying description. We introduce here extended L-ensembles, and show that all DPPs are extended L-ensembles (and vice-versa). Extended L-ensembles have very simple likelihood functions, contain L-ensembles and projection DPPs as special cases. From a theoretical standpoint, they fix some pathologies in the usual formalism of DPPs, for instance the fact that projection DPPs are not L-ensembles. From a practical standpoint, they extend the set of kernel functions that may be used to define DPPs: we show that conditional positive definite kernels are good candidates for defining DPPs, including DPPs that need no spatial scale parameter. Finally, extended L-ensembles are based on so-called ``saddle-point matrices'', and we prove an extension of the Cauchy-Binet theorem for such matrices that may be of independent interest.